Formula for Pi

Just curious as to whether there is an exact formula to calculate pi using algebra.....Disregarding c/d or any approximation such as 22/7.

Is there any relation to primes and while im asking is there any formula for finding an Nth term prime. I.E. a pattern showing where a prime will crop up.

Im currently trying to formulate an equation that finds pi using only numbers 1-4. Where a = 1, b = 2, c = 3 and d = 4;

something like; (d/c) * ((d/c) + (c/b) * ((d/c) / (c/b)))); Makes something within 2 decimal places, obviously this is just random and at some point must happen but im curious whether a formula along these lines already exist and can prove pi in purely mathematical terms?

thx

P.s Im thinking of writing a program that calculates every value for 1-3 combo, so 1/3, 2/3, 1/2 and then jumbles these answers up so that it does 3/2 + 1/3 * (2*3)/3; etc etc. Has anyone tried this??

The rational numbers are closed under addition, multiplication, subtraction and division, so no finite combination of rational numbers using those operations will give you an exact result for pi, which is not a rational number.

Edited by John, 6 March 2013 - 08:22 AM.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule

Just curious as to whether there is an exact formula to calculate pi using algebra.....Disregarding c/d or any approximation such as 22/7.

...

Im currently trying to formulate an equation that finds pi using only numbers 1-4. Where a = 1, b = 2, c = 3 and d = 4;

Won't work unfortunately. Since is irrational, it can't be derived by doing arithmetic on integers. Approximations can be achieved, but you'll never make an expression like such equivalent to the true value of .

Though you can't derive it this way, there are certainly other ways to represent it, many of them quite elegant and beautiful.

Is there any relation to primes and while im asking is there any formula for finding an Nth term prime. I.E. a pattern showing where a prime will crop up.

By the way, there is a prime-counting function denoted . It just tells you how many prime numbers there are at and below the given value. The notation is only due from the Greek letter however, and is not a reference to the constant .

Won't work unfortunately. Since is irrational, it can't be derived by doing arithmetic on integers. Approximations can be achieved, but you'll never make an expression like such equivalent to the true value of .

Though you can't derive it this way, there are certainly other ways to represent it, many of them quite elegant and beautiful.

By the way, there is a prime-counting function denoted . It just tells you how many prime numbers there are at and below the given value. The notation is only due from the Greek letter however, and is not a reference to the constant .

Very interesting stuff, ill give it a read tomorrow when my heads clear.

Purely out of curiosity the numbers that are used as C/D = pi are presumably rational (but not integer)?, if so, can't any of them be created using just the number 1?? ((1 + 1 + 1) / (1) * (1+1)) etc etc...

Given C/D = pi, the fact that pi is irrational means C or D is irrational. If C and D are both rational, then since rational numbers are closed under division, C/D cannot equal pi.

Just to be clear, when we say a set is closed under some operation, we mean the result of applying that operation to members of the set results in another member of the set. Thus, what I'm saying is that given any two rational numbers, dividing one by the other yields another rational number. Therefore, pi, being irrational, cannot be the result of dividing one rational number by another.

Edited by John, 7 March 2013 - 08:32 AM.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule

Given C/D = pi, the fact that pi is irrational means C or D is irrational. If C and D are both rational, then since rational numbers are closed under division, C/D cannot equal pi.

Just to be clear, when we say a set is closed under some operation, we mean the result of applying that operation to members of the set results in another member of the set. Thus, what I'm saying is that given any two rational numbers, dividing one by the other yields another rational number. Therefore, pi, being irrational, cannot be the result of dividing one rational number by another.

Does this imply that the nature of a circle is irrational?,also where in physical reality does a pure circle exist? im trying to conceptualise how and why a circle doesnt fit into a perfect mathematical system, as i currently understand it, it has something to do with the way in which we represent 3-d space in a 2-d mathematical geometry.

irrational in maths and when talking about numbers has a clearly defined meaning - an irrational number is any real number that cannot be expressed as a ratio of two integers. If you want to talk about circles you can say that the ratio of the radius to the circumference is irrational.

Circles are mathematical entities - they exist in maths; they are the locus of all points on a plane equidistant from a point. Many things in nature are circular - the equator is a pretty fine example. Anything that tends to increase volume for a fixed surface area will begin to look spherical - and its cross sections will be circular. The existence of pure circles - ideal circles - relates to the notion of platonic ideals

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A little learning is a dangerous thing; drink deep, or taste not the Pierian spring:
there shallow draughts intoxicate the brain, and drinking largely sobers us again.- Alexander Pope

I'm not sure what you mean by the "nature" of a circle. The ratio of a circle's diameter to its circumference is pi, which is irrational, and the area of a circle is equal to its radius multiplied by pi squared, so in that sense, I guess it is.

As for circles existing in reality, you'd be hard-pressed to find anything perfectly circular, especially since we can't measure things like length with infinite precision.

I'm also not certain what you mean by "perfect mathematical system," but assuming such a thing exists, irrational numbers aren't necessarily excluded. While we usually think of "irrational" as meaning "unreasonable," in mathematics the "rational numbers" are simply those that can be expressed as a ratio of two integers. Irrational numbers cannot be written as the ratio of two integers, hence their name.

As for the last bit, keep in mind that the sphere (a 3D object) is similar in that finding its volume and surface area, based on its radius, also involves pi.

Edited by John, 7 March 2013 - 04:36 PM.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule

I'm not sure what you mean by the "nature" of a circle. The ratio of a circle's diameter to its circumference is pi, which is irrational, and the area of a circle is equal to its radius multiplied by pi squared, so in that sense, I guess it is.

As for circles existing in reality, you'd be hard-pressed to find anything perfectly circular, especially since we can't measure things like length with infinite precision.

I'm also not certain what you mean by "perfect mathematical system," but assuming such a thing exists, irrational numbers aren't necessarily excluded. While we usually think of "irrational" as meaning "unreasonable," in mathematics the "rational numbers" are simply those that can be expressed as a ratio of two integers. Irrational numbers cannot be written as the ratio of two integers, hence their name.

As for the last bit, keep in mind that the sphere (a 3D object) is similar in that finding its volume and surface area, based on its radius, also involves pi.

Okay firstly as previously stated for pi to be irrational one of the 2 variables in finding the ratio must also be irrational this leads me to believe a circle must always have an irrational diameter or circumference which means that before even evaluating the ratio between the 2, 1 is already irrational.

A perfect mathematical system is one where everything is a measurement of an accumulation of the smallest possible unit. You cant divide by this unit as it would make no logical sense, if 1 cent is the smallest possible unit, 3 people cannot share a physical cent so theres no logic in dividing it by anything else. I suppose a system where remainders accumulate over time and then become new numbers, not forced into being divided. In theory getting rid of infinity i suppose. In a system where infinity doesnt exist, pi has no meaning.

Though its only assigned for human purposes, if one cycle of a sine wave is skewed, could it be a perfect circle?? Or could some exact point relative to the magnetic force create a perfect circle??

irrational in maths and when talking about numbers has a clearly defined meaning - an irrational number is any real number that cannot be expressed as a ratio of two integers. If you want to talk about circles you can say that the ratio of the radius to the circumference is irrational.

Circles are mathematical entities - they exist in maths; they are the locus of all points on a plane equidistant from a point. Many things in nature are circular - the equator is a pretty fine example. Anything that tends to increase volume for a fixed surface area will begin to look spherical - and its cross sections will be circular. The existence of pure circles - ideal circles - relates to the notion of platonic ideals

How do we prove that a circle has 360 degree's? any idea what happens if we drop 90 and use 270 as a measurement? having 3 sets of right angles instead of 4

Okay firstly as previously stated for pi to be irrational one of the 2 variables in finding the ratio must also be irrational this leads me to believe a circle must always have an irrational diameter or circumference which means that before even evaluating the ratio between the 2, 1 is already irrational.

Correct.

A perfect mathematical system is one where everything is a measurement of an accumulation of the smallest possible unit. You cant divide by this unit as it would make no logical sense, if 1 cent is the smallest possible unit, 3 people cannot share a physical cent so theres no logic in dividing it by anything else. I suppose a system where remainders accumulate over time and then become new numbers, not forced into being divided. In theory getting rid of infinity i suppose. In a system where infinity doesnt exist, pi has no meaning.

This seems an odd definition of perfection for a mathematical system. I'm not sure how a lack of infinity renders pi meaningless. Assuming that by "getting rid of infinity," you mean nothing can be infinite (including the number of digits after a decimal point), the ratio of a circle's circumference to its diameter would still be approximately pi.

Though its only assigned for human purposes, if one cycle of a sine wave is skewed, could it be a perfect circle?? Or could some exact point relative to the magnetic force create a perfect circle??

You'd need to ask someone better educated in physics to be more sure, but my understanding is that it would be impossible for us to verify that anything physical is in fact a perfect circle, due to limitations in measuring equipment at small scales and eventually due to the uncertainty principle at quantum scales.

How do we prove that a circle has 360 degree's? any idea what happens if we drop 90 and use 270 as a measurement? having 3 sets of right angles instead of 4

The degree is defined as 1/360 of a full rotation, therefore a full rotation (i.e. a circle) is 360 degrees. Dropping 90 degrees would result in three quarters of a circle.

You could redefine the degree to be 1/270 of a rotation, I suppose, but then we'd just say a right angle is 67.5 degrees.

Edited by John, 7 March 2013 - 08:17 PM.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule

Nowhere. Everything in nature that is close to circular isn't quite a circle.

That's irrelevant, however. While scientists and engineers use mathematics to describe reality, mathematics itself is not bound by reality. That a "pure circle" does not exist in physical reality doesn't matter to mathematicians.

Even if a "pure circle" did exist in physical reality, it's circumference would not be exactly 2πr because physical reality is not Euclidean. Space-time is instead, as far as we know, pseudo-Riemannian. Rhetorical question: Did special and general relativity disprove Euclidean geometry? The answer is no. While those theories did show that Euclidean geometry isn't a perfect descriptor of our universe, they did not disprove Euclidean geometry itself. Mathematics is not bound by reality.

im trying to conceptualise how and why a circle doesnt fit into a perfect mathematical system, as i currently understand it it has something to do with the way in which we represent 3-d space in a 2-d mathematical geometry.

That's nonsense. Jumping ahead to your next post where you expanded upon this,

A perfect mathematical system is one where everything is a measurement of an accumulation of the smallest possible unit. You cant divide by this unit as it would make no logical sense, if 1 cent is the smallest possible unit, 3 people cannot share a physical cent so theres no logic in dividing it by anything else.

Okay firstly as previously stated for pi to be irrational one of the 2 variables in finding the ratio must also be irrational this leads me to believe a circle must always have an irrational diameter or circumference which means that before even evaluating the ratio between the 2, 1 is already irrational.

Yes, one or both of the radius and circumference must be irrational.

How do we prove that a circle has 360 degree's? any idea what happens if we drop 90 and use 270 as a measurement? having 3 sets of right angles instead of 4

The degree is defined as 1/360th of a full circle. You can't prove or disprove a definition.

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This is modern software; there's no useful manual. After all, changing how everything works every six months or so is more important than helping people use the features you have.

Nowhere. Everything in nature that is close to circular isn't quite a circle.

That's irrelevant, however. While scientists and engineers use mathematics to describe reality, mathematics itself is not bound by reality. That a "pure circle" does not exist in physical reality doesn't matter to mathematicians.

Even if a "pure circle" did exist in physical reality, it's circumference would not be exactly 2πr because physical reality is not Euclidean. Space-time is instead, as far as we know, pseudo-Riemannian. Rhetorical question: Did special and general relativity disprove Euclidean geometry? The answer is no. While those theories did show that Euclidean geometry isn't a perfect descriptor of our universe, they did not disprove Euclidean geometry itself. Mathematics is not bound by reality.

That's nonsense. Jumping ahead to your next post where you expanded upon this,

The degree is defined as 1/360th of a full circle. You can't prove or disprove a definition.

Space-time or GR contradicts the 5th euclidean postulate, which makes it wrong.

Engineers use mathematics for real purposes, though maths itself isnt bound to reality the only maths we actually need to know, understand and discover are bound to reality, all the rest are pointless probabilities that didnt and wont exist, else again they are bound to reality (so all maths that isnt relative to reality is void of purpose (also if we create an equation or formula mathematically that isnt directly related to reality it could be very detrimental if used in certain physical situations like the hadron collider or a nuclear submarine)).

I'm currently working on a thesis that deals with this perfect mathematical system i purpose, it has a direct relation to shapes and topological math aswell as the nature of circles and time. Perhaps once ive finished i'll drop it on here and you can pick it apart, but for now positive integers will do to show how circles dont work within the nature of reality.

Just a little side question im confused with...from origin on the double positive part of a polar circle graph, how do we calculate the co-ordinates for the first half of a sine wave in terms of degree's? as in were working with a 90 degree right angle origin and were trying to account for or calculate the points of a semi circle, which is 180 degrees?

if the answer is splitting the angle into .5's how small can we split angles?

if the answer is relative to one axis representing 3-d (some measurement of energy (mass, speed, force etc)) and the other representing time (so this right angle is a representation of 4-d) then why are the vectors connecting the vertices curved and not 2-d lines? (as the crow flies so to speak)

I still cant quite comprehend how a circle fits into reality, even based on the fact that only having 3 digits of pi is a good enough approximation of reality to use for engineering purposes, how can it be infinite?? there must surely be some cut of point where a fractal pattern emerges? such as an infinite regression based on recursion (a single base unit).

Space-time or GR contradicts the 5th euclidean postulate, which makes it wrong.

Though I'm not sure whether you mean GR is wrong or Euclidean geometry is wrong, this is probably incorrect in either case, depending on what exactly you mean by "wrong."

Engineers use mathematics for real purposes, though maths itself isnt bound to reality the only maths we actually need to know, understand and discover are bound to reality, all the rest are pointless probabilities that didnt and wont exist, else again they are bound to reality (so all maths that isnt relative to reality is void of purpose (also if we create an equation or formula mathematically that isnt directly related to reality it could be very detrimental if used in certain physical situations like the hadron collider or a nuclear submarine)).

Don't let pure mathematicians hear you say that. Also, mathematics that seems to have no application now may find applications in the future. For an example, see the history of number theory.

I'm currently working on a thesis that deals with this perfect mathematical system i purpose, it has a direct relation to shapes and topological math aswell as the nature of circles and time. Perhaps once ive finished i'll drop it on here and you can pick it apart, but for now positive integers will do to show how circles dont work within the nature of reality.

I won't speak for D H, but I will say "perfect" still strikes me as odd terminology for what seemingly amounts to manipulating the natural numbers.

Just a little side question im confused with...from origin on the double positive part of a polar circle graph, how do we calculate the co-ordinates for the first half of a sine wave in terms of degree's? as in were working with a 90 degree right angle origin and were trying to account for or calculate the points of a semi circle, which is 180 degrees?

if the answer is splitting the angle into .5's how small can we split angles?

if the answer is relative to one axis representing 3-d (some measurement of energy (mass, speed, force etc)) and the other representing time (so this right angle is a representation of 4-d) then why are the vectors connecting the vertices curved and not 2-d lines? (as the crow flies so to speak)

I'm not sure what exactly you're asking here. Maybe I'm just too tired. If no one else answers, perhaps you could clarify your question a bit.

I still cant quite comprehend how a circle fits into reality, even based on the fact that only having 3 digits of pi is a good enough approximation of reality to use for engineering purposes, how can it be infinite?? there must surely be some cut of point where a fractal pattern emerges? such as an infinite regression based on recursion (a single base unit).

A circle is simply the set of all points at some distance from a given central point. It's not a physical object, just a geometric shape. It fits into reality insofar as certain objects or regions are approximately (but not perfectly) circular.

I'm not sure what you're asking with regards to the "infinite" question. If you're asking about pi's decimal representation, then it's infinitely long because pi is an irrational number, one proof of which can be found here.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule

Though I'm not sure whether you mean GR is wrong or Euclidean geometry is wrong, this is probably incorrect in either case, depending on what exactly you mean by "wrong."

As for GR and euclidean geometry, either or both are wrong, in some form they must be. If they dont fit together and one contradicts the other, then something must be missing.

Don't let pure mathematicians hear you say that. Also, mathematics that seems to have no application now may find applications in the future. For an example, see the history of number theory.

Sorry; to clarify, this accounts for every law which makes reality, every piece of math we use or can be used. Anything that has no practical application need not exist (not everything in math reflects reality, that which doesnt, will not).

I won't speak for D H, but I will say "perfect" still strikes me as odd terminology for what seemingly amounts to manipulating the natural numbers.

A finite system is a suffice definition of perfect, until i can demonstrate fully what i mean.

I'm not sure what exactly you're asking here. Maybe I'm just too tired. If no one else answers, perhaps you could clarify your question a bit.

Im to tired now, I'll get some sleep and make a diagram explaining the question tomorrow.

A circle is simply the set of all points at some distance from a given central point. It's not a physical object, just a geometric shape. It fits into reality insofar as certain objects or regions are approximately (but not perfectly) circular.

I'm not sure what you're asking with regards to the "infinite" question. If you're asking about pi's decimal representation, then it's infinitely long because pi is an irrational number, one proof of which can be found here.

Yes both the concept of irrational (infinite) and circle geometry are currently beyond my reach If i cant comprehend infinity i must come up with another solution to fill its place hence the use of a *perfect* system.

As for GR and euclidean geometry, either or both are wrong, in some form they must be. If they dont fit together and one contradicts the other, then something must be missing.

They're both valid within their respective domains of applicability. I suppose something is missing with respect to general relativity in the sense that we haven't yet united it with quantum mechanics. As for Euclidean geometry, as far as I know nothing's missing. It's a mathematical system based on intuitive axioms, and I believe it's complete and probably consistent. A consequence of general relativity is that real space isn't Euclidean, but Euclidean geometry is still a good approximation given certain conditions (namely, I think, in regions where gravitational effects are fairly weak).

Sorry; to clarify, this accounts for every law which makes reality, every piece of math we use or can be used. Anything that has no practical application need not exist (not everything in math reflects reality, that which doesnt, will not).

No worries, my response here was mostly tongue in cheek. However, though I currently intend to go into applied mathematics, I don't agree that pure mathematics and its results are worthless or need not exist.

Yes both the concept of irrational (infinite) and circle geometry are currently beyond my reach If i cant comprehend infinity i must come up with another solution to fill its place hence the use of a *perfect* system.

If the decimal representation of pi were of finite length, then one could just multiply pi by a sufficient power of 10 to get an integer P, which could then be written as P/1, a ratio of two integers. Since pi is irrational, it cannot be expressed as a ratio of two integers, which means its decimal representation must be infinitely long.

Based on some of what you've said so far, you might find finitism somewhat pleasing. It's not exactly the same as what you're talking about here, but you might enjoy looking into some of its ideas.

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"The capsule bore precious cargo: a thimble-full of particles gathered from the tail of a comet--comets, of course, frozen bodies of ice and dust formed over 4.6 billion years ago, or created 6000 years ago depending on whether or not you're wrong." -The Daily Show, on the Stardust space capsule